Simha/Sinha/Siha means lion in Indian languages, which is straightforward enough. But the controversial part is the suffix -la. For most of its history the popular etymology of Simhala/Sinhala/Sihala has been interpreted as “lion-killer”. The Mahavamsa, a 6th century CE Pali chronicle, in narrating the origin mythology of the Sinhala dynasty, explains that a north Indian prince called Sihabahu captured and killed his own father, a marauding lion, thus his descendants led by prince Vijaya who established themselves in Lanka came to be called Sihala. In Sanskrit lā means to grasp, seize or take and since the story about Sihabahu describes him as seizing the lion and since lā and la are a homonym, the later works such as Vamsatthappakasini and Saddaniti explain Sihala as meaning seizing and killing the lion.
However, this seems to be a later folk etymology rather than the original meaning of Sihala. In Sanskrit, the nominal suffix -la expresses characterization or possession, similar to the English adjective combining form -like as in lifelike. For example, the Sanskrit words parṇala means full of leaves or leafy and śmaśrula means bearded. Hence, Sihala probably meant possessing characteristics of a lion or lion-like.
This Sri Lankan lion mythology may have been a later reworking of the earlier Buddhist stories about the merchant Simhala from the Indian mainland probably going back to the BCE. The Valāhassa Jātaka narrates a story about a merchant and his caravan of five hundred traders being shipwrecked on an island, being seduced by man-eating demonesses (yakkhinis or rākṣasīs). The Divyāvadāna, dated to the 2nd century CE, narrates the same story about a caravan leader called Simhala who derives his name from his father, Simha. Simhala and his men are shipwrecked on Tamradvipa (island of copper) inhabited by cannibalistic yakkhinis. Simhala flees from the island and gets back to his own country (Simhakalpa) in the mainland where he is elected king. Later on he leads a military expedition to the island, vanquishing the yakkhinis and establishing a kingdom there. The island came to called Simhaladvipa after King Simhala’s name.
The local chronicle, Mahavamsa, shares with these stories the name Simhala and a group of north Indians sailing to Lanka and taking the native demon women of the island, the yakkhinis, as their wives. One major difference, however, is that in the Indian stories there's no account of capturing and slaying a lion, hence Simhala in the Indian context means either lion-like or son of a (man called) Simha. Interestingly, I-Tsing, a Chinese Buddhist monk in the late 7th century CE, refers to Lanka as “the island of the Son of the Lion”.
Faxian, a Chinese Buddhist monk in the late 4th century CE, describes in his travelogue how the “Lion country” was formed: Initially it was only inhabited by spirits such as the nagas who were later subdued by the Buddha. Its pearls and precious stones attracted merchants from various countries and when people from various countries had heard about how great the island was many came there and settled in it. There is no story about killing or seizing a lion is mentioned here. The Dipavamsa, the prequel to the Mahavamsa, composed in the 4th century CE, doesn't mention the lion killing story either, although it recognizes the father of Sihabahu as an actual lion. The Bahiranidana composed perhaps a century later doesn’t mention the incident either. Hence, the story appears to be a later addition which had emerged by the 6th century CE.
In the early 7th century CE, another Chinese pilgrim, Xuanzang, notes that the island was formerly known as Ratnadvipa (island of gems) due to the precious gems found there and that it was occupied by evil spirits. He goes onto narrate, though with some modifications, the story of the Mahavamsa, of a princess being taken by a lion and cohabiting with her, resulting in the birth to two children, a son and a daughter. The major differences in Xuanzang's version are that the princess was born in a Southern Indian kingdom in contrast to the Mahavamsa's Vanga (Bengal) and the son gets banished to Lanka for killing his own father whereas in the Mahavamsa the son (Sihabahu) establishes a kingdom in India, marries his own sister who gives birth to prince Vijaya and it's Vijaya who gets banished to Lanka for being a troublemaker. In Xuanzang's version it's the son of the lion exiled in Lanka who establishes the kingdom which, on the account of his capturing the lion, came to be called Simhala. Interestingly, in the same account of the island, Xuanzang also mentions the Indian story about the merchant Simhala and his caravan of 500 merchants sailing to Ratnadvipa. Hence, Xuanzang recognizes that two versions of the Sinhala origin stories existed.
If there's any truth behind the Indian story of the merchant Simhala, it appears to be older and more believable than the local tradition about Vijaya and his grandfather being an actual lion. If not, the whole Simhala story may have first originated in the mainland as a mere fable and only later on the islanders adopted it. The word Sihala does not appear anywhere in the early history of the island, not even in its numerous Prakrit Brahmi inscriptions going back to the 4th century BCE, until the 4th century CE work Dipavamsa. This chronicle makes a single mention of the term: that lankadipa (island of Lanka) was called sihala on account of the lion. It also lists the former names of the island as well as recent ones such as lankadipa and tambapanni but excludes sihala. Its sequel Mahavamsa too mentions the term sihala only once. The Bahiranidana, focusing on the early history of the island up to the 1st century BCE, uses the term tambapanni instead of sihala. In his 3rd century BCE edicts, Ashoka too uses the term tambapanni as the name of the island. This may imply the term Sihala had not yet been in vogue in the island and was only later on being adopted perhaps from the mainland.
Today Sihala refers to both a language and an ethnie, but in its earliest usages it was mostly a toponym referring to the island itself. So, when the old sources mention sihaladipa, they do not mean island of the Sinhalese ethnie as we understand it today but lion-like island.
Perhaps there was a real merchant named Simha in Western India who had a son that was named after him as Simhala; like his father, Simhala became a merchant and sailed to Ratnadvipa seeking precious gems. He may have settled in the island and established a kingdom and gave his name to it, Simhala.
In sum, Sinhala could mean lion-like.
On a side note, the forms heḷa and eḷu are actually mutations of Pali sīhaḷa in the old Sinhala language (9th century CE), with the initial h being a substitute for s, e.g., Pali sīhaḷadipa appears as heḷa divi; and the form heḷa became eḷu as the initial h is also usually dropped, e.g., hand in Sanskrit is hasta which becomes ata. Rather than these variants being “pure” Sinhala words as certain nationalists contend, they are “corruptions” of classical Sanskrit, e.g., Sanskrit Śrī Laṃkā appears as siri laka in old Sinhala. Hence, Siṃhala>Sīhaḷa>heḷa>eḷu.
As for the reinterpretation of sīhala as sīhalē, meaning “lion blood” since lē in Sinhala means blood, it’s a modern fanciful derivation based on a near homophone but unknown to the earliest tradition.