All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
There is nothing wrong with naming a child Al-Arqam, as this was the name of one of the Companions, may Allah be pleased with him. Had this name included a forbidden meaning, then the Prophet would have commanded him to change it, as he did with some other Companions.
The linguists stated that Al-Arqam is a name for a type of snakes, but this name has several other meanings that do not include a forbidden meaning; among which is that Ar-Raqm means putting patterns or engraving; it is said “Raqama Ath-Thawb” when one put patterns on a piece of clothing. It also refers to inscribing; it is said “Raqama Al-Kitaab” if one clarifies its letters and puts dots and vowels on the letters.
On the other hand, not every animal name is prohibited to be named with; Shaykh Bakr Abu Zayd said, “It is disliked to be named with the names of animals that are well-known for despicable qualities, such as being named Hanash (snake), Himaar (donkey), Qirdaan (a bird called cattle egret [Bubulcus ibis]), Kalb (dog)... The Arabs named their children with such names because of the good meaning that they include; a dog is always alert and provides benefit; a donkey is patient and bears hardship, and so forth.”
Allah knows best.