If your significant other is not your spouse and not a dependent on your return, NO, you cannot claim a medical expense deduction for them.
If this is your spouse or they are a dependent on your return, then yes, you can include lyme disease treatment as an itemized medical expense.
Itemized expenses include mortgage interest, gambling losses up to winnings, charitable contributions, state and local taxes up to $10,000, medical expenses in excess of 7.5% of your AGI and casualty and losses in excess of 10% of you AGI with the first $100 not counting towards the loss. All medical expenses are only deductible for the amount that is over 7.5% of your AGI. This means if your AGI is $50,000, then the amount that is over $3,750 is deductible.
Then your total itemized expenses would need to be greater than your standard deduction below in order to benefit from your expenses.